Problem 3. Let A be a complex nxn (n ≥2) normal matrix. Suppose that λ = 1 is an eigenvalueof A satisfy |μ< 1. Proveof A of (algebraic) multiplicity one and that all other eigenvaluesthat the sequence of powers of A converges entrywise to a nonzero matrix of the form xx*, i.e.,lim Ak = xx* for some nonzero à E Cn.k-400

Answered: Image /qna-images/answer/60a9804f-2ab4-449a-a3e8-e36d81a6f3a7.jpg

Answered: Problem 3. Let A be a complex nxn (n…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1.5.3 (a) If the eigenvalues of the n x n matrix A are λ > λ > 13... λ = 0 show that the eigenvalue λ, can be found by applying the power method to the matrix kl - A, where is the identity matrix and k > 2₁.
31. (a) Let A be an n × n-symmetric matrix. Diagonalize A to show that x • Ax 2 || x || ² is greater than or equal to the smallest eigenvalue of A for all x # 0 in R". (b) Show that the quadratic form Q₁(x) = x Ax is coercive if and only if A is positive definite. (c) Conclude from (b) that if f(x) = a + b x + 1x. Ax is any quadratic function where a = R, be R₁ and A is an n - n-symmetric matrix, then f(x) is coercive if and only if A is positive definite.
in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1
(7) Suppose Q is a matrix with orthonormal columns. (a) Show that multiplication by Q preserves length; that is, ||QT|| = ||F|| for any i. (b) Suppose 7 is an eigenvector for Q; that is, a nonzero vector satisfying Qữ = ca for some scalar c. Show that c=±1.
6. Consider the matrix A = Compute the eigenvalues of A, and the corresponding eigenspaces.
A = To 1 0 101 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, where the A, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A =Σ/x)(x| = Σ\x(x)". i) Show that if the eigenvalues are real then A, as defined through the above expansion, is Hermitian. ii) Using the result for A show that the Nx N identity matrix can be written as 1=Σx'(x¹)¹. i=1
4. Let At be a matrix that depends on r and is given by At t 3 -2 1 (a) For which te R matrix At has no real eigenvalues for At? (b) For which t E R matrix At has exactly one real distinct eigenvalue? Give that eigenvalue. (c) For which te R matrix At has two different real eigenvalues? Give one such eigenvalue as a function of t.
4. A real, symmetric 3-by-3 matrix M has eigenvalues X₁ = 4, X₂ 1 vector corresponding to the eigenvalue X₁ = 4 is v₁ -1 0 [4] -2 to the eigenvalue X₂ = 6 is v₂ = 6 and X3 = -3. An eigen- and an eigenvector corresponding What is M?
part 1 real analysis
4. The eigenvalues of a matrix A € Cnxn are usually not computed in practice by finding the zeros of the characteristic polynomial. One way to approximate the eigenvector corre- sponding to the eigenvalue of a diagonalizable matrix A with largest absolute value is power iteration, Axi ||Axi||2 Xi+1 i = 0, 1, 2, ..., where o C is some initial guess. Approximation for the eigenvalue of A with largest absolute value is computed as Hi = R(A, x₂), i = 0, 1, 2, ... (1) The intuition behind power iteration is that repeated multiplication by A turns the vector slowly towards the eigenvector corresponding to the eigenvalue of A with largest absolute value (unless xo is orthogonal to it). Let now [2 1 0] A = 1 2 1 0 1 2 Choose o [1,0,0]¹. Compute an approximation for the eigenvalue of A with largest absolute value Xamax (A) and the corresponding eigenvector by using power iteration. Plot the error Xamax (A) — μ| as a function of the index i = 0, 1,..., 10.
7. Let the 2 x 2 matrix A have real, distinct eigenvalues and μ. Suppose that an eigenvector of A is (1,0)" and an eigenvector of u is (-1,1)". Sketch the phase portraits of is x = Ax for the following cases: (1) 0 0. وچستان ستان ج
22. Let P be an idempotent matrix. Show that the only eigenvalues of P are 1 Suppose that Px = Ax, x + 0.] 0 and 1 = 1. [Hint:

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS